Course page

Course page

Instructor: Franca Hoffmann

Course Description: This course gives an overview of different mathematical tools to analyze partial differential equations encountered across mathematics, biology, physics, engineering and social science. We will look at the main properties of different classes of linear and nonlinear PDEs and the behavior of their solutions using tools from functional analysis and calculus of variations with an emphasis on applications. We will focus on representative models from different areas which may include: Lotka-Volterra equations, the Fokker-Planck equation, the Boltzmann equation, the Fisher/KPP equation, Burger's equation, swarming models, models for opinion dynamics, bio-mathematics models for cell movement and bacterial chemotaxis (Patlack- Keller-Segel model), SIR models from epidemiology, predator-prey systems, chemical reactions, enzymatic reactions. The above list is flexible and depends on the audience. If you are interested in this course, feel free to contact the instructor with (types of) models you would like to study.

We will study the theories involved in understanding the following key concepts:

• Between different scales: Micro-Meso-Macro
• Agent-based models, kinetic equations and corresponding macroscopic descriptions
• Mean Field Limits and Hydrodynamic Scalings
• Derivatives and Theory of Distributions
• Transport Equations

Instructor: Joel A. Tropp

Course Description: This course offers a rigorous introduction to probability and stochastic processes. Emphasis is placed on the interaction between inequalities and limit theorems, as well as contemporary applications in computing and mathematical sciences. Topics include probability measures, random variables and expectation, independence, concentration inequalities, distances between probability measures, modes of convergence, laws of large numbers and the central limit theorem, Gaussian and Poisson approximation, conditional expectation and conditional distributions, filtrations, and discrete-time martingales.

Instructor: Houman Owhadi

Course Description: Syllabus:

• Gaussian vectors
• Gaussian processes, measures and fields
• Gaussian process regression
• Statistical numerical approximation
• Kernel methods and Reproducing Kernel Hilbert Spaces
• Kernel PCA
• Kernel LDA
• Kernel mean embedding
• Branching (Galton-Watson) Processes
• Poisson (Point) Processes
• Dirichlet processes

Instructor: Andrew Stuart

Course Description: Rationale: At the end of the course you will have covered:

• Precise statements of mathematical models for properties of matter, at different scales: quantum, molecular and continuum
• Formal understanding of inter-relations between these different models
• Elementary manipulations of the equations
• Implementation of a numerical method for one of the models present
• Presentation of material to peers

Syllabus: Tentative topics are the following:

• Tensor algebra and calculus (2 lectures, Week 1)
• Continuum Mechanics of Fluids and Solids (2 lectures, Week 1)
• Quantum Mechanics (4 lectures, Week 2)
• Molecular Dynamics (4 lectures, Week 3)
• Kinetic Theory (4 lectures, Week 4)